Benford’s Law for Coefficients of Newforms

# Benford’s Law for Coefficients of Newforms

Marie Jameson Jesse Thorner  and  Lynnelle Ye Marie Jameson, Department of Mathematics, University of Tennessee, Knoxville, TN 37996 Jesse Thorner, Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 30322 Lynnelle Ye, Department of Mathematics, Harvard University, Cambridge, MA 02138
###### Abstract.

Let be a newform of even weight on without complex multiplication. Let denote the set of all primes. We prove that the sequence does not satisfy Benford’s Law in any integer base . However, given a base and a string of digits in base , the set

 Aλf(b,S):={\textup$p$prime:thefirstdigitsof$λf(p)$inbase$b$aregivenby$S$}

has logarithmic density equal to . Thus follows Benford’s Law with respect to logarithmic density. Both results rely on the now-proven Sato-Tate Conjecture.

## 1. Introduction and Statement of Results

In 1881, astronomer Simon Newcomb [SN] made the observation that certain pages of logarithm tables were much more worn than others. Users of the tables referenced logarithms whose leading digit is 1 more frequently than other logarithms, contrary to the naive expectation that all of the logarithms would be referenced uniformly. In 1938, Benford [Benford] made a similar observation for a variety of sequences.

This bias, now known as Benford’s Law, is given as follows. Let denote the set of positive integers, and let be an infinite subset. Fix an integer base and a string of digits in base . For a given function , define

 (1.1) Ag(b,S):={\textup$i∈N$:thefirstdigitsof$g(i)$inbase$b$aregivenby$S$}.

We define the arithmetic density of within by

 (1.2) δI(Ag(b,S))=limx→∞#{i≤x:i∈I∩Ag(b,S)}#{i≤x:i∈I}.

We say that the sequence satisfies Benford’s Law, or that is Benford, if for any integer base and any string of digits in base ,

 (1.3) δI(Ag(b,S))=logb(1+S−1).

It is easy to show [Diaconis] that is Benford if and only if the set is equidistributed modulo 1 for each base (setting if ). For some general surveys on Benford’s Law, we refer the reader to [Hill1, Hill2, MTB, Raimi].

Stirling’s approximation of in conjunction with standard equidistribution results quickly yields that is a Benford sequence. However, if for any fixed , then is not a Benford sequence, for

 limsupn∈N|nlog(|g(n+1)/g(n)|)|<∞,

contradicting equidistribution of modulo 1. Taking , one sees that the positive integers are not Benford.

Much real world data, including lengths of rivers, populations of nations, and heights of skyscrapers exhibit behavior which is suggestive of Benford’s Law (when restricted to base 10). There are also several settings in which Benford’s law arises that are of arithmetic interest. In [BBH], several dynamical systems such as linearly-dominated systems and non-autonomous dynamical systems are shown to be Benford. Benford’s Law is proven for distributions of values of -functions [KM] and the problem [KM, LS]. In [ARS], the image of the partition function is shown to be Benford, as well as the coefficients of an infinite class of modular forms with poles.

While the positive integers are not Benford, we can say even more. Specifically, if we take , the limits defining and for any do not exist. For example, one easily sees that

 liminfx→∞#{i≤x:i∈N∩Ag(10,1)}#{i≤x:i∈N}=19,limsupx→∞#{i≤x:i∈N∩Ag(10,1)}#{i≤x:i∈N}=59.

However, if we change our notion of density, the first digits of the integers still satisfy the distribution in Benford’s law. We define the logarithmic density of in by

 (1.4) ˜δI(Ag(b,S))=limx→∞∑i≤xi∈I∩Ag(b,S)i−1∑i≤xi∈Ii−1.

With this modified notion of density, we have that exists and equals for any base and any string in base [Duncan]. In light of this fact, we say that a sequence is logarithmically Benford if

 (1.5) ˜δI(Ag(b,S))=logb(1+S−1)

for any base and any string in base . We note that if a set has an arithmetic density, then it also has a logarithmic density, and the two densities are equal. Thus all Benford sequences are logarithmically Benford.

###### Remark.

Logarithmic density is closely related to Dirichlet density, which is ubiquitous in number theory. For integer sequences, logarithmic density and Dirichlet density have equivalent definitions; this (and much more) is proven in Part 3 of [Tenenbaum]. For a discussion on Dirichlet density in the context of the prime number theorem for arithmetic progressions or the Chebotarev density theorem, see Chapter 7 of [Neukirch].

Since the -th prime is asymptotically equal to by the prime number theorem, one sees that the primes are not Benford. However, Whitney [Whitney] proved that for , we have for any string in base 10. The fact that the primes are logarithmically Benford follows easily from Whitney’s proof. Serre briefly discusses this problem for the primes in Chapter 4, Section 5 of [arithmetic].

In this paper, we consider sequences given by Fourier coefficients of certain modular forms without complex multiplication (see [Ono]). Specifically, let

 (1.6) f(z)=∞∑n=1λf(n)qn∈S\textupnewk(Γ0(N)),q=e2πiz

be a newform (i.e., a holomorphic cuspidal normalized Hecke eigenform) of even weight and trivial nebentypus on that does not have complex multiplication. The Fourier coefficients of such a newform will be real. We consider sets of the form

 Aλf(b,S)={\textup$n∈N$:thefirstdigitsof$λf(n)$inbase$b$aregivenby$S$}.

One important example of the newforms under our consideration is the weight 12 newform on given by

 Δ(z)=q∞∏n=1(1−qn)24=∞∑n=1τ(n)qn,

where is the Ramanujan tau function. Consider the following table.

If were Benford, then we would have . If we only consider the above table, then it seems to be the case that exists and might very well equal . However, the plot in Figure 1 indicates that this conclusion is very far from the truth.

It turns out to be the case that the arithmetic density does not exist. However, does exist, and it equals . More generally, we prove the two following results.

###### Theorem 1.

Let be a newform of even weight without complex multiplication. The arithmetic density does not exist for any integer base , and the arithmetic density does not exist. Thus the sequence is not Benford.

###### Remark.

For a string in base , the method used in the proof of Theorem 1 can be modified to show that there are infinitely many primes such that begins with . Since this fact is also a direct consequence of Theorem 2, we omit the details.

###### Theorem 2.

Let be a newform of even weight without complex multiplication. Let be a given integer base, and let be an initial string of digits in base . We have . Thus is logarithmically Benford.

### Acknowledgements

The authors thank Ken Ono and the anonymous referee for their comments and Ken Ono for suggesting this project. The authors used Maple 18, Mathematica 9, and SAGE for the numerical computations and plots.

## 2. The Sato-Tate Conjecture

Let

 f(z)=∞∑n=1λf(n)qn∈S\textupnewk(Γ0(N))

be a newform of even weight and trivial nebentypus on . The Fourier coefficients will lie in the ring of integers of a totally real number field. By Deligne’s proof of the Weil conjectures, we have that for every prime ,

 |λf(p)|≤2pk−12.

Thus there exists satisfying

 λf(p)=2pk−12cosθp.

Around 1960, Sato and Tate studied the sequence as varies through the primes when is the newform associated to an elliptic curve without complex multiplication. All such newforms have weight , and if is prime, then

 λf(p)=p+1−#E(Fp),|λf(p)|≤2√p.

where is the number of -rational points on . Thus is the normalized error in approximating with . Sato and Tate conjectured a distribution for the sequence , and this conjecture was later generalized to a much larger class of newforms. This conjecture, which we now state, was proven by Barnet-Lamb, Geraghty, Harris, and Taylor [sato-tate].

###### Theorem 3 (The Sato-Tate Conjecture).

Let be a newform of even weight without complex multiplication. The sequence is equidistributed in the interval with respect to the measure

 dμST=2π√1−t2 dt.

In other words, if is a subinterval and we define

 πf,I(x)=#{p≤x:cosθp∈I},

then as ,

 πf,I(x)∼μST(I)π(x).

The following immediate corollary of the Sato-Tate Conjecture plays an important role in the proof of Theorem 2.

###### Corollary 1.

Let be a newform of even weight without complex multiplication. Let be an interval. As , we have

 ∑p≤xcosθp∈Ip−1∼μST(I)∑p≤xp−1.

## 3. Proof of theorem 1

To prove Theorem 1, we use the Sato-Tate Conjecture to construct many large intervals on which the proportion of primes for which has leading digit in a given base differs from the Benford expectation. (For base , we use the leading digits 10 because all nonzero real numbers have leading digit 1 in their base 2 expansion.) This shows that is not Benford in any base. To do this, we first state a lemma about the Sato-Tate measures of certain intervals.

###### Lemma 1.

Fix and let be a sufficiently large positive integer. For , set

 Id,1(c)=⋃j∈Z[b−jdbc−2,2b−jdbc−1]∩[0,1].

Then

 2μST(I2,1(c))−2μST(I1,1(c))>140.
###### Proof of Lemma 1.

If , then

 1b

Thus

 I1,1(c)=⋃j≥−c+1[b−jbc−2,2b−jbc−1]=⋃m≥0[b−1−m1−2b−c,2b−1−m1−b−c]

and

 I2,1(c) =[bc2bc−2,1]∪⋃j>−c[b−j2bc−2,2b−j2bc−1] =[12−2b−c,1]∪⋃m>0[b−m2−2b−c,2b−m2−b−c].

For all , we have

 limc→∞(μST([12−2b−c,1])−μST([b−11−2b−c,2b−11−b−c])) = 2μST([1/2,1])−2μST([b−1,2b−1])>140.

Thus for all sufficiently large , we have

 2μST([12−2b−c,1])−2μST([b−11−2b−c,2b−11−b−c])>140

Similarly, we have

 2μST([b−m2−2b−c,2b−m2−b−c])−2μST([b−1−m1−2b−c,2b−1−m1−b−c])>0

for all and the result follows. ∎

Now we may prove Theorem 1.

###### Proof of Theorem 1.

Let be a newform as in the statement of the theorem and let . In order prove that the arithmetic density does not exist, it suffices to show that for some fixed , the value of

 limn→∞#{αβn≤p<γβn:p∈Aλf(b,1)}#{αβn≤p<γβn}

varies with different choices of values of and with .

 I1,S(c) =⋃j∈Z[Sb−jbc−2,(S+1)b−jbc−1]∩[0,1], I2,S(c) =⋃j∈Z[Sb−j2bc−2,(S+1)b−j2bc−1]∩[0,1].

Fix , and let be a sufficiently large positive integer so that Lemma 1 holds and

 μST([0,1]−∪b−1S=1Id,S(c))<ϵ/4

for . Let , , and . We consider the primes such that

 (3.1) (bc−2)bn≤2pk−12<(bc−1)bn.

Note that if is bounded as in (3.1) and then

 |λf(p)|∈[Sbn−j,(S+1)bn−j)

for some ; that is, its first digits are given by . By letting be sufficiently large and setting , the Sato-Tate Conjecture implies that

 (3.2) ∣∣ ∣∣limn→∞#{α1βn≤p<γ1βn:p∈Aλf(b,1)}#{α1βn≤p<γ1βn}−2μST(I1,1(c))∣∣ ∣∣<ϵ2.

Similarly, by letting and , we find that

 (3.3) ∣∣ ∣∣limn→∞#{α2βn≤p<γ2βn:p∈Aλf(b,1)}#{α2βn≤p<γ2βn}−2μST(I2,1(c))∣∣ ∣∣<ϵ2.

Now, suppose to the contrary that exists. It follows from (3.2) and (3.3) that

 |2μST(I1,1)(c)−2μST(I2,1)(c)|<ϵ<140,

which contradicts Lemma 1. The theorem now follows for bases . For , one arrives at the same conclusion as for by comparing for in Lemma 1. The computations for are essentially the same. ∎

## 4. Proof of Theorem 2

For this section, will always denote a prime. Let be a newform satisfying the hypotheses of Theorem 2. Let be a base, and let be a string of digits in base . By the definition of a logarithmically Benford sequence and the estimate

 (4.1) ∑p≤xp−1∼loglogx,

a proof of Theorem 2 will follow from proving that as ,

 (4.2) ∑p≤xp∈Aλf(b,S)p−1∼logb(1+S−1)loglogx.

This will be a consequence of the following key lemma.

###### Lemma 2.

Let be a newform of even weight without complex multiplication. Let be a given base, let be an initial string of digits in base , and let be an integer. As , we have

 (1+o(1))(logb(1+S−1)−log(1+ℓ−1))loglogx ≤ ∑p≤xp∈Aλf(b,S)|cosθp|>ℓ−1p−1 ≤ (1+o(1))(logb(1+S−1)+log(1+ℓ−1))loglogx+2loglogℓ.
###### Proof.

We prove the upper bound; the lower bound is proven similarly. Writing , we first observe that

 ∑p≤xp∈Aλf(b,S)|cosθp|>ℓ−1p−1 =∞∑t=−∞ℓ2−1∑i=ℓ∑p≤xS⋅bt≤|λf(p)|<(S+1)⋅btiℓ2<|cosθp|≤i+1ℓ2p−1 ≤∞∑t=−∞ℓ2−1∑i=ℓ∑p≤x(Sℓ2i+1bt)2k−1≤p≤((S+1)ℓ2ibt)2k−1iℓ2<|cosθp|≤i+1ℓ2p−1.

To bound the contribution when , all of the primes in the sum are at most ; since , we have

 −1∑t=−∞ℓ2−1∑i=ℓ∑p≤x(Sℓ2i+1bt)2k−1≤p≤((S+1)ℓ2ibt)2k−1iℓ2<|cosθp|≤i+1ℓ2p−1≤∑p≤((S+1)ℓb)2k−1p−1≤2loglogℓ.

To bound the contribution when , fix . Recall that we may write . If , then Corollary 1 implies that

 ∑p≤xS⋅bt≤|λf(p)|<(S+1)⋅btiℓ2<|cosθp|≤i+1ℓ2p−1 ≤ ∑(Sℓ2i+1bt)2k−1≤p≤((S+1)ℓ2ibt)2k−1iℓ2<|cosθp|≤i+1ℓ2p−1 = 2(1+o(1))μST([iℓ2,i+1ℓ2])∑(Sℓ2i+1bt)2k−1≤p≤((S+1)ℓ2ibt)2k−1p−1 = 2(1+o(1))μST([iℓ2,i+1ℓ2])log⎛⎜⎝logb(ℓ2i)+logb(S+1)+tlogb(ℓ2i+1)+logb(S)+t⎞⎟⎠.

Setting and , we have for any large that

 N∑t=0∑(Sℓ2i+1bt)2k−1≤p≤((S+1)ℓ2ibt)2k−1p−1∼logN∏t=0B1+tB0+t.

Switching the order of summation, we obtain the inequality

 ∞∑t=0ℓ2−1∑i=ℓ∑p≤xS⋅bt≤|λf(p)|<(S+1)⋅btiℓ2<|cosθp|≤i+1ℓ2p−1 ≤ ℓ2−1∑i=ℓ2μ([iℓ2,i+1ℓ2])(1+o(1))∞∑t=0∑(Sℓ2i+1bt)2k−1≤p≤min⎧⎪⎨⎪⎩((S+1)ℓ2ibt)2k−1,x⎫⎪⎬⎪⎭p−1 ≤ ℓ2−1∑i=ℓ2μ([iℓ2,i+1ℓ2])(1+o(1))log⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝∏0≤t≤logb⎛⎜⎝ixk−12(S+1)ℓ2⎞⎟⎠B1+tB0+t⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠+2loglogℓ.

Using Euler’s formula for the Gamma function

 Γ(z)=limn→∞n!nz∏ni=0(z+i),

we find that the contribution from is at most

 ℓ2−1∑i=ℓ2μ([iℓ2,i+1ℓ2])(1+o(1))(logΓ(B0)Γ(B1)+(B1−B0)loglogx) ≤ (1+o(1))(logb(1+ℓ−1)+logb(1+S−1))loglogx.

This proves the claimed upper bound. Using the inequality

 ∑p≤xS⋅bt≤2pk−12|cosθp|<(S+1)⋅btiℓ<|cosθp|≤i+1ℓp−1≥∑(Sℓibt)2k−1≤p≤((S+1)ℓi+1bt)2k−1iℓ<|cosθp|≤i+1ℓp−1

for , the lower bound is proven similarly. ∎

###### Proof of Theorem 2.

Let , let be an integer, and let . If we write with , then

 (4.3) ∑p≤xp∈Af(b,S)p−1=∑p≤xp∈Aλf(b,S)|cosθp|≤ℓ−1p−1+∑p≤xp∈Aλf(b,S)|cosθp|>ℓ−1p−1.

By Corollary 1, the first term is at most

 (4.4) ∑p≤x0≤|cosθp|≤ℓ−1p−1=(2+o(1))μST([0,ℓ−1])loglogx.

Using Lemma 2, we now have

 (1+o(1))(logb(1+S−1)−logb(1+ℓ−1))loglogx ≤ ∑p≤xp∈Af(b,S)p−1 ≤ (1+o(1))(logb(1+S−1)+logb(1+ℓ−1)+2μST([0,ℓ−1]))loglogx+2loglogℓ.

Thus

 (1+o(1))(logb(1+S−1)−ϵ)loglogx ≤ ∑p≤xp∈Af(b,S)p−1 ≤ (1+o(1))(logb(1+S−1)+9log(b)ϵ)loglogx.

Letting , we obtain (4.2), as desired. ∎

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